In 6-dimensional geometry, the **1 _{22}** polytope is a uniform polytope, constructed from the E

_{6}group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V

_{72}(for its 72 vertices).

## Contents

- 122 polytope
- Alternate names
- Construction
- Related complex polyhedron
- Related polytopes and honeycomb
- Geometric folding
- Tessellations
- Rectified 122 polytope
- Images
- References

Its Coxeter symbol is **1 _{22}**, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 1

_{22}, construcated by positions points on the elements of 1

_{22}. The

**rectified 1**is constructed by points at the mid-edges of the

_{22}**1**. The

_{22}**birectified 1**is constructed by points at the triangle face centers of the

_{22}**1**.

_{22}These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

## 1_22 polytope

The 1_22 polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E_{6}.

## Alternate names

**Pentacontatetra-peton**(Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)

## Construction

It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram, .

Removing the node on either of 2-length branches leaves the 5-demicube, 1_{31}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 0_{22}, .

## Related complex polyhedron

The regular complex polyhedron _{3}{3}_{3}{4}_{2}, , in
*1 _{22}* polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is

_{3}[3]

_{3}[4]

_{2}, order 1296. It has a half-symmetry quasiregular construction as , as a rectification of the Hessian polyhedron, .

## Related polytopes and honeycomb

Along with the semiregular polytope, **2 _{21}**, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

## Geometric folding

The **1 _{22}** is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to

**1**in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 1

_{22}_{22}.

## Tessellations

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, **2 _{22}**, .

## Rectified 1_22 polytope

The **rectified 1 _{22}** polytope (also called

**0**) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).

_{221}## Alternate names

_{21}polytope

*Ram*) - rectified 54-facetted polypeton (Jonathan Bowers)

## Construction

Its construction is based on the E_{6} group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Removing the ring on the short branch leaves the birectified 5-simplex, .

Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: **t _{2}(2_{11})**, .

The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×{}, .

## Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

## Alternate names

_{22}polytope

## Construction

Its construction is based on the E_{6} group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

## Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

## Alternate names

_{21}

## Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

## Alternate names

_{21}